{"id":"https://openalex.org/W2130462033","doi":"https://doi.org/10.1137/110827041","title":"Intrinsic Stationarity for Vector Quantization: Foundation of Dual Quantization","display_name":"Intrinsic Stationarity for Vector Quantization: Foundation of Dual Quantization","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W2130462033","doi":"https://doi.org/10.1137/110827041","mag":"2130462033"},"language":"en","primary_location":{"id":"doi:10.1137/110827041","is_oa":false,"landing_page_url":"https://doi.org/10.1137/110827041","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1010.4642","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5090083053","display_name":"Gilles Pag\u00e8s","orcid":"https://orcid.org/0000-0001-6487-3079"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Gilles Pag\u00e8s","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5088444316","display_name":"Benedikt Wilbertz","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Benedikt Wilbertz","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5090083053"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.8354,"has_fulltext":false,"cited_by_count":13,"citation_normalized_percentile":{"value":0.76799649,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"50","issue":"2","first_page":"747","last_page":"780"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10901","display_name":"Advanced Data Compression Techniques","score":0.4505000114440918,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10901","display_name":"Advanced Data Compression Techniques","score":0.4505000114440918,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.0575999990105629,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.02199999988079071,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7847551107406616},{"id":"https://openalex.org/keywords/delaunay-triangulation","display_name":"Delaunay triangulation","score":0.6570462584495544},{"id":"https://openalex.org/keywords/quantization","display_name":"Quantization (signal processing)","score":0.6347155570983887},{"id":"https://openalex.org/keywords/vector-quantization","display_name":"Vector quantization","score":0.6199517846107483},{"id":"https://openalex.org/keywords/simplex","display_name":"Simplex","score":0.6174148321151733},{"id":"https://openalex.org/keywords/voronoi-diagram","display_name":"Voronoi diagram","score":0.6019085049629211},{"id":"https://openalex.org/keywords/stochastic-quantization","display_name":"Stochastic quantization","score":0.5786858201026917},{"id":"https://openalex.org/keywords/grid","display_name":"Grid","score":0.5710762143135071},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.4310646653175354},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.37545979022979736},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.34902575612068176},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.29265454411506653},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.19406676292419434},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13021808862686157},{"id":"https://openalex.org/keywords/path-integral-formulation","display_name":"Path integral formulation","score":0.08628949522972107}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7847551107406616},{"id":"https://openalex.org/C68010082","wikidata":"https://www.wikidata.org/wiki/Q192445","display_name":"Delaunay triangulation","level":2,"score":0.6570462584495544},{"id":"https://openalex.org/C28855332","wikidata":"https://www.wikidata.org/wiki/Q198099","display_name":"Quantization (signal processing)","level":2,"score":0.6347155570983887},{"id":"https://openalex.org/C199833920","wikidata":"https://www.wikidata.org/wiki/Q612536","display_name":"Vector quantization","level":2,"score":0.6199517846107483},{"id":"https://openalex.org/C62438384","wikidata":"https://www.wikidata.org/wiki/Q331350","display_name":"Simplex","level":2,"score":0.6174148321151733},{"id":"https://openalex.org/C24881265","wikidata":"https://www.wikidata.org/wiki/Q757267","display_name":"Voronoi diagram","level":2,"score":0.6019085049629211},{"id":"https://openalex.org/C2780288589","wikidata":"https://www.wikidata.org/wiki/Q7617833","display_name":"Stochastic quantization","level":4,"score":0.5786858201026917},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.5710762143135071},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4310646653175354},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37545979022979736},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.34902575612068176},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.29265454411506653},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.19406676292419434},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13021808862686157},{"id":"https://openalex.org/C154018700","wikidata":"https://www.wikidata.org/wiki/Q898323","display_name":"Path integral formulation","level":3,"score":0.08628949522972107},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/110827041","is_oa":false,"landing_page_url":"https://doi.org/10.1137/110827041","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1010.4642","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1010.4642","pdf_url":"https://arxiv.org/pdf/1010.4642","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:HAL:hal-00528485v2","is_oa":true,"landing_page_url":"https://hal.science/hal-00528485","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Numerical Analysis, 2012, 50, pp.747-780. ⟨10.1137/110827041⟩","raw_type":"Journal articles"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1010.4642","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1010.4642","pdf_url":"https://arxiv.org/pdf/1010.4642","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W1576475658","https://openalex.org/W1811570306","https://openalex.org/W1970978299","https://openalex.org/W1979253219","https://openalex.org/W2004158166","https://openalex.org/W2012595274","https://openalex.org/W2048370823","https://openalex.org/W2082176355","https://openalex.org/W2087197119","https://openalex.org/W2093125922","https://openalex.org/W2102197848","https://openalex.org/W2146636127","https://openalex.org/W3099273337","https://openalex.org/W3123843963"],"related_works":["https://openalex.org/W2354869996","https://openalex.org/W1481755670","https://openalex.org/W2359751290","https://openalex.org/W2206755844","https://openalex.org/W2371448287","https://openalex.org/W2358109229","https://openalex.org/W2096247527","https://openalex.org/W431872879","https://openalex.org/W2360695371","https://openalex.org/W2355968994"],"abstract_inverted_index":{"We":[0,97],"develop":[1],"a":[2,42,55,74,100,132],"new":[3,122],"approach":[4,123],"to":[5,19,51,104,131],"vector":[6],"quantization,":[7,21],"which":[8,46],"guarantees":[9],"an":[10,87],"intrinsic":[11],"stationarity":[12],"property":[13],"that":[14,67,127],"also":[15,98],"holds,":[16],"in":[17],"contrast":[18],"regular":[20],"for":[22,38,90,110,137],"nonoptimal":[23],"quantization":[24,40,95],"grids.":[25],"This":[26],"goal":[27],"is":[28,65,124],"achieved":[29],"by":[30,41],"replacing":[31],"the":[32,48,52,60,78,84,125,142,145],"usual":[33],"nearest":[34],"neighbor":[35],"projection":[36],"operator":[37],"Voronoi":[39],"random":[43,49],"splitting":[44],"operator,":[45],"maps":[47],"source":[50],"vertices":[53],"of":[54,57,77,86,120,141,144],"triangle":[56],"d-simplex.":[58],"In":[59],"quadratic":[61],"Euclidean":[62],"case,":[63],"it":[64,128],"shown":[66],"these":[68],"triangles":[69],"or":[70,93],"d-simplices":[71],"make":[72],"up":[73],"Delaunay":[75],"triangulation":[76],"underlying":[79,146],"grid.":[80,147],"Furthermore,":[81],"we":[82],"prove":[83],"existence":[85],"optimal":[88,107],"grid":[89],"this":[91,121],"Delaunay,":[92],"dual,":[94],"procedure.":[96],"provide":[99],"stochastic":[101],"optimization":[102],"method":[103],"compute":[105],"such":[106],"grids,":[108],"here":[109],"higher":[111],"dimensional":[112],"uniform":[113],"and":[114],"normal":[115],"distributions.":[116],"A":[117],"crucial":[118],"feature":[119],"fact":[126],"automatically":[129],"leads":[130],"second":[133],"order":[134],"quadrature":[135],"formula":[136],"computing":[138],"expectations,":[139],"regardless":[140],"optimality":[143]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2013,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2016-06-24T00:00:00"}